MA3ANT-Analytic Number Theory

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:6
Terms in which taught: Spring term module
Pre-requisites: MA2RCA Real and Complex Analysis or MA2RA2 Real Analysis II and MA2CA1 Complex Analysis I or MA1RA2NU Real Analysis II MA2CANU Complex Analysis and
Non-modular pre-requisites:
Co-requisites: MA3Z7 Number Theory
Modules excluded: MA4ANT Analytic Number Theory
Current from: 2022/3

Module Convenor: Dr Jani Virtanen
Email: j.a.virtanen@reading.ac.uk

Type of module:

Summary module description:

This module introduces complex analytic techniques in the theory of numbers.Ìý

Aims:

To introduce complex analytic techniques in the theory of numbers.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

• use real analytic methods in the theory of arithmetical functions;


• find analytic continuations of functions;


• formulate and solve problems using Dirichlet series;


• appreciate the proof of the prime number theorem and the link with the Riemann zeta function.

Additional outcomes:

Outline content:

This module uses techniques from complex analysis to study problems in number theory. First we develop some necessary real and complex analysis, in particular the notion of analytic continuation. After studying the Gamma function, we develop the theory of Dirichlet series of a complex variable. This will be followed by a study of the Riemann zeta function – a central object in analytic number theory – and its connection to the distribution of prime numbers.

Brief description of teaching and learning methods:

Lectures supported by problem sheets.

Contact hours:

Ìý	Autumn	Spring	Summer
Lectures		20
Guided independent study:		80
Ìý	Ìý	Ìý	Ìý
Total hours by term		100
Ìý	Ìý	Ìý	Ìý
Total hours for module

Summative Assessment Methods:

Method	Percentage
Written exam	100

Summative assessment- Examinations:

2 hours.

The examination for this module will require a narrowly defined time window and is likely to be held in a dedicated exam venue.

Summative assessment- Coursework and in-class tests:

Formative assessment methods:

Problem sheets.

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

The University policy statement on penalties for late submission can be found at: /cqsd/-/media/project/functions/cqsd/documents/cqsd-old-site-documents/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:

A mark of 40% overall.

Reassessment arrangements:

One examination paper of 2 hours duration in August/September.

Additional Costs (specified where applicable):

Last updated: 22 September 2022